Finite Group Databasebeta

← #556347

Subgroups — #556347

Order 512.

2678 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2625C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3C2×#2296C211yesmaximal, characteristic
4#14627C214yesmaximal
5#14343C211yesmaximal, characteristic
6#14328C212yesmaximal, complemented
7#14326C212yesmaximal, complemented
8#14318C212yesmaximal, complemented
9#14314C212yesmaximal, complemented
10#14264C211yesmaximal, characteristic
11#14258C211yesmaximal, characteristic
12#14202C214yesmaximal
13#14045C212yesmaximal, complemented
14#14042C212yesmaximal, complemented
15#14021C212yesmaximal, complemented
16#14017C212yesmaximal, complemented
17C4×#380C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2625C4116yescomplemented
24C2×#400C2212yes—
25C2×#398C2212yes—
26C2×#397C2214yes—
27C4×C22⋊Q8—28no—
28#2296—28no—
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#387C2212yes—
40C2×#387C2214yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2212yes—
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2212yes—
54#1445C2212yes—
55#1445C2212yes—
56#1444C2214yes—
57#1444C2214yes—
58#1444C2212yes—
59#1443C2214yes—
60#1443C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1442C2212yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2212yes—
67#1441C2212yes—
68#1440C2214yes—
69#1440C2214yes—
70#1439C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81#1434C2214yes—
82#1434C2214yes—
83#1434C2212yes—
84C22⋊#95C2212yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8C2×C418yes—
93C2×C4×Q8—24no—
94C2×C4×Q8—44no—
95C2×#117—24no—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117—44no—
99C2×#117C2×C418yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C42—44no—
125C22×C42D418yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2314yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#400C2×C4116yescomplemented
136#398C2×C4116yescomplemented
137#397C2×C4132yes—
138#391—24no—
139#391—44no—
140#391C2×C4116yescomplemented
141#389—24no—
142#389—24no—
143#389—24no—
144#389—24no—
145#389—24no—
146#389—44no—
147#389C2×C4116yescomplemented
148#387—28no—
149#387—48no—
150#385—28no—
151#385—28no—
152#385—28no—
153#380—24no—
154#380—24no—
155#380C2×C418yes—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4—24no—
160C4×C4⋊C4—44no—
161C4×C22⋊C4—28no—
162C4×C22⋊C4—44no—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C25C2411yescharacteristic
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8C22×C414yes—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—42no—
203C23×C4—41no—
204C23×C4—41no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4C22×C414yes—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4#4114yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C22⋊Q8—232no—
229C4×Q8—416no—
230#117—232no—
231#117—416no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4C22×C414yes—
276C2×C4⋊C4C22×C414yes—
277C2×C4⋊C4C22×C414yes—
278C2×C4⋊C4C22×C418yes—
279C2×C4⋊C4C22×C418yes—
280C2×C4⋊C4C22×C4116yes—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4C22×C418yes—
296C2×C22⋊C4C22×C418yes—
297C2×C22⋊C4C22×C418yes—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—28no—
310C2×C42—416no—
311C2×C42—44no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42C22×C418yes—
317C2×C42C2×D414yes—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320C2×C42C2×D414yes—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—28no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95C22×C418yes—
346#95C22×C4116yes—
347#95C22×C4116yes—
348#95C22×C4116yes—
349C24—21no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—21no—
354C24—21no—
355C24—22no—
356C24—21no—
357C24C23×C412yes—
358C24C22×D412yes—
359C24C2×#4112yes—
360C24C2511yescharacteristic
361C2×Q8—28no—
362C2×Q8—216no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—48no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—216no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—216no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4C4×D4116yes—
455C22×C4C23×C414yes—
456C22×C4C23×C418yes—
457C22×C4C23×C414yes—
458C22×C4C23×C414yes—
459C22×C4C23×C414yes—
460C22×C4C23×C412yes—
461C22×C4C23×C412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C4H(5,𝔽2)14yes—
472C22×C42−1+414yes—
473C22×C42−1+412yes—
474C22×C42−1+414yes—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—216no—
479C4⋊C4—216no—
480C4⋊C4—416no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—88no—
493C22⋊C4—216no—
494C22⋊C4—216no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C42—216no—
500C42—28no—
501C42—28no—
502C42—44no—
503C42—44no—
504C42—48no—
505C42—48no—
506C42—48no—
507C42—88no—
508C23—24no—
509C23—22no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—22no—
520C23—24no—
521C23—22no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23C2×C4×D414yes—
531C23C24×C411yescharacteristic
532C23C23×D411yescharacteristic
533C23C22×#4111yescharacteristic
534C23C2×H(5,𝔽2)12yes—
535C23C2×H(5,𝔽2)11yescharacteristic
536C23C2×2−1+412yes—
537C23C2×2−1+412yes—
538C23C2×2−1+411yescharacteristic
539C23#58511yescharacteristic
540C23#58511yescharacteristic
541C23#58512yes—
542Q8—416no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—232no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—416no—
577C2×C4—416no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4C2×C4×D414yes—
633C2×C4C2×C4×D414yes—
634C2×C4#52018yes—
635C2×C4#53414yes—
636C2×C4#53614yes—
637C2×C4#53614yes—
638C2×C4#53714yes—
639C2×C4#53918yes—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—42no—
651C22C22×C4×D411yescharacteristic
652C22C2×#52012yes—
653C22C4×H(5,𝔽2)11yescharacteristic
654C22C4×2−1+412yes—
655C22C4×2−1+411yescharacteristic
656C22C2×#53411yescharacteristic
657C22C2×#53611yescharacteristic
658C22C2×#53611yescharacteristic
659C22C2×#53711yescharacteristic
660C22#344911yescharacteristic
661C22#344911yescharacteristic
662C22C2×#53912yes—
663C22#346011yescharacteristic
664C22#346112yes—
665C22#346111yescharacteristic
666C22#353612yes—
667C22#353812yes—
668C22#353912yes—
669C22#354012yes—
670C22#355412yes—
671C22#355512yes—
672C22#355512yes—
673C22#355512yes—
674C4—24no—
675C4—24no—
676C4—24no—
677C4—44no—
678C4—44no—
679C4—44no—
680C4—44no—
681C4—42no—
682C4—42no—
683C4—48no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—84no—
688C4—82no—
689C4#232218yes—
690C2—44no—
691C2C2×#232211yescharacteristic
692C2C4×#53611yescharacteristic
693C2C4×#53711yescharacteristic
694C2#3441912yes—
695C2#3442012yes—
696C2#3744211yescharacteristic
697C2#3744511yescharacteristic
698C2#3752412yes—
699C2#3752612yes—
700C2#3762211yescharacteristic
701C2#3763511yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. #14328C2semidirect
7. #14326C2semidirect
8. #14318C2semidirect
9. #14314C2semidirect
13. #14045C2semidirect
14. #14042C2semidirect
15. #14021C2semidirect
16. #14017C2semidirect
23. #2625C4semidirect
135. #400C2×C4semidirect
136. #398C2×C4semidirect
140. #391C2×C4semidirect
147. #389C2×C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#26252
C2×C4×C22⋊Q82
C2×#22962
#146272
#143432
#143282
#143262
#143182
#143142
#142642
#142582
#142022
#140452
#140422
#140212
#140172
C4×#3802